Induced Disjoint Paths and Connected Subgraphs for $H$-Free Graphs

Abstract

Paths P 1 , . . . , P k in a graph G = (V, E) are mutually induced if any two distinct P i and P j have neither common vertices nor adjacent vertices. The Induced Disjoint Paths problem is to decide if a graph G with k pairs of specified vertices (si, ti) contains k mutually induced paths P i such that each P i starts from si and ends at ti. This is a classical graph problem that is NP-complete even for k = 2. We introduce a natural generalization, Induced Disjoint Connected Subgraphs: instead of connecting pairs of terminals, we must connect sets of terminals. We give almost-complete dichotomies of the computational complexity of both problems for H-free graphs, that is, graphs that do not contain some fixed graph H as an induced subgraph. Finally, we give a complete classification of the complexity of the second problem if the number k of terminal sets is fixed, that is, not part of the input.